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I’m thinking…
1. R1e2+ Kd1 2. Kxc3 Kc1 3. Re1#
1. R1e2+ Kc1 2. Rd3 Kb1 3. Rd1#
1. R1e2+ Kb1 2. Rf3 Ka1 3. Rf1#
1. R1e2+ Kb1 2. Rf3 Kc1 3. Rf1#
1. R1e2+ Kc1 2. Rd3 Kb1 3. Rd1#
1.R1e2? doesn’t seem to work due to,
1…Kc1
2.Rd3 c2! and is not mate in 3.
1. Rd1
if 1… Kb2
2 Rxc3 Ka2
3 Rc2#
if 1… Kxd1
2 Kxc3 Kc1
3 Re1 #
Gegga
1. Rd1!
A)
1. … Kxd1
2. Kxc3 Kc1
3. Re1#
B)
1. …Kb2
2. Rxc3 Ka2
3. Rc2#
Like the previous two people pointed out, 1. Rd1! is the only way to checkmate in 3.
1. R1e2+ takes 5 moves after 1…Kb1.
1. R1e2+ takes 5 moves after 1…Kb1
1.R1e2 Kb1?
2.Rxc3 Ka1
3.Rc1#
Of course black would respond differently,
1.R1e2 Kc1! and this is mate in four.
That was rather easy Checkmate in 3.